IODS course project
Paavo Laitinen
Introduction to Open Data Science - Course Project
About the project
Write a short description about the course and add a link to your GitHub repository here. This is an R Markdown (.Rmd) file so you should use R Markdown syntax.
# This is a so-called "R chunk" where you can write R code.
date()## [1] "Mon Dec 5 09:34:07 2022"The text continues here.
I hope everything is now installed correctly. Learning Git/Github seems useful and it is something that should prove to be useful in the future. I am a bit nervous about using R as I’m not very familar with it but we’ll see how this goes.
Thoughts about Exercise set 1
R really seems like a good tool for building graphs and plots and the logic seems pretty straightforward. However, it will take time to learn the syntax and all the commands. It is inevitable that learning new software feels at first like an avalanche of information, but this tutorial for R was nice in that it teaches one thing at a time.
Chapter 2) Regression and model validation
Describe the work you have done this week and summarize your learning.
- Describe your work and results clearly.
- Assume the reader has an introductory course level understanding of writing and reading R code as well as statistical methods.
- Assume the reader has no previous knowledge of your data or the more advanced methods you are using.
date()## [1] "Mon Dec 5 09:34:07 2022"1) Reading data
lrn14 <- read.table("learning2014.csv", sep=",", header=TRUE)
# dimensions of the data
dim(lrn14)## [1] 166 7# structure of the data
str(lrn14)## 'data.frame': 166 obs. of 7 variables:
## $ gender : chr "F" "M" "F" "M" ...
## $ age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ attitude: int 37 31 25 35 37 38 35 29 38 21 ...
## $ deep : num 3.58 2.92 3.5 3.5 3.67 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 ...
## $ points : int 25 12 24 10 22 21 21 31 24 26 ...This is survey data from 2014 of Approaches to learning.This is a subset of original learning2014 data, from which 7 variables were picked:gender, age, attitude, deep, stra, surf and points. ‘Deep’(deep learning), ‘stra’ (strategic learning) and ‘surf’ (superficial learning) have been combined from their related items. You can find more information about the dataset from learning2014
2) Overview of data
library(ggplot2)
library(GGally)## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2p <- ggpairs(lrn14, mapping = aes(col = gender, alpha = 0.3), lower = list(combo = wrap("facethist", bins = 20)))
p
Looking at the data, there are more females than males. Age is the only variable that is clearly skewed (to the left), while other variables are more evenly distributed. Srongest correlations are found between deep learning and surface learning (-0.32), attitude and point (0.44). Additionally, surface leaning was correlated with strategic learning (-0.16) and attitude (-0.18). All these aforementioned correlations were in same direction in both sexes.
3) and 4) Linear regression model
model_1 <- lm(points ~ attitude + stra + surf, data = lrn14)
summary(model_1)##
## Call:
## lm(formula = points ~ attitude + stra + surf, data = lrn14)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.1550 -3.4346 0.5156 3.6401 10.8952
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.01711 3.68375 2.991 0.00322 **
## attitude 0.33952 0.05741 5.913 1.93e-08 ***
## stra 0.85313 0.54159 1.575 0.11716
## surf -0.58607 0.80138 -0.731 0.46563
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.296 on 162 degrees of freedom
## Multiple R-squared: 0.2074, Adjusted R-squared: 0.1927
## F-statistic: 14.13 on 3 and 162 DF, p-value: 3.156e-08model_2 <- lm(points ~ attitude, data = lrn14)
summary(model_2)##
## Call:
## lm(formula = points ~ attitude, data = lrn14)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.9763 -3.2119 0.4339 4.1534 10.6645
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.63715 1.83035 6.358 1.95e-09 ***
## attitude 0.35255 0.05674 6.214 4.12e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared: 0.1906, Adjusted R-squared: 0.1856
## F-statistic: 38.61 on 1 and 164 DF, p-value: 4.119e-09Initially, I chose attitude, strategic learning and surface learning as explanatory variables for points in linear regression model as they were the variables showing highest correlation with points. However, attitude was the only statistically significant predictor of points (Coeff 0.34, p-value <0.001), while the model explained 20% of variation in points (Adjusted R squared). After removing non significant variables from the model, attitude predicted points with Coeff 0.35 (p-value <0.001), while the model explained 19% of variation in points. Taken together, based on this model, attitude was the best predictor of points: better attitude predicted more points.
5) Diagnostic plots
Residuals vs Fitted values, Normal QQ-plot and Residuals vs Leverage
par(mfrow = c(2,2))
plot(model_2, which = c(1, 2, 5))
The main assumptions of Linear regression model are:
* linear relationship between response value and eplanatory value
* errors (residuals) have constant variance across all values of eplanatory variable
* errors are independent of each other
* errors have a normal distribution
Residuals vs fitted should not show a pattern where distribution of residuals varies along fitted values. In the figure above residuals are nicely evenly distributed.
QQ-plot should show a line if residuals are normally distributed. Figure above shows that the data is pretty much following the line excluding few outliers.
Residuals vs leverage should not show points outside Cook’s distance, which holds for the figure above.
Taken together, the assumptions of linear regression model hold true for the current model.
Chapter 3) Logistic regrsession
date()## [1] "Mon Dec 5 09:34:14 2022"Reading data
The data consists of student performance data in two Portuguese schools, collected by reports and questionnaires. It consists of two datasets from two different subjects: math and Portuguese. These datasets were combined and the variables of the combined dataset are printed below. In addition, two variables were calculated: alc_use is the average alcohol use from weekdays and weekends and high_use tells is true when alc_use is over 2
More information (including variable information) and the original dataset can be found from UCI_machine_learning_repository
alc <- read.csv("C:/Users/labpaavo/IODS-project/data/alc.csv")
colnames(alc)## [1] "school" "sex" "age" "address" "famsize"
## [6] "Pstatus" "Medu" "Fedu" "Mjob" "Fjob"
## [11] "reason" "guardian" "traveltime" "studytime" "schoolsup"
## [16] "famsup" "activities" "nursery" "higher" "internet"
## [21] "romantic" "famrel" "freetime" "goout" "Dalc"
## [26] "Walc" "health" "failures" "paid" "absences"
## [31] "G1" "G2" "G3" "alc_use" "high_use"Variables of interest in relation to alcohol consumption and hypotheses
For the variables of interest in relation to high/low alcohol consumption I have chosen: sex, Pstatus, studytime and absences. The hypotheses are that:
- being a male predicts high alcohol consumption
- parents living apart predicts high alcohol consumption
- low studytime predicts high alcohol consumption
- high absences predicts high alcohol consumption
Exploring the chosen variables
Numerically and graphically explore the distributions of your chosen variables and their relationships with alcohol consumption (use for example cross-tabulations, bar plots and box plots). Comment on your findings and compare the results of your exploration to your previously stated hypotheses. (0-5 points)
library(descr)
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(ggplot2)
my_variables = c("high_use", "sex", "Pstatus", "studytime", "absences")
my_data <- alc[my_variables]
summary(my_data)## high_use sex Pstatus studytime
## Mode :logical Length:370 Length:370 Min. :1.000
## FALSE:259 Class :character Class :character 1st Qu.:1.000
## TRUE :111 Mode :character Mode :character Median :2.000
## Mean :2.043
## 3rd Qu.:2.000
## Max. :4.000
## absences
## Min. : 0.000
## 1st Qu.: 1.000
## Median : 3.000
## Mean : 4.511
## 3rd Qu.: 6.000
## Max. :45.000CrossTable(my_data$high_use, my_data$sex)## Cell Contents
## |-------------------------|
## | N |
## | Chi-square contribution |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
## =========================================
## my_data$sex
## my_data$high_use F M Total
## -----------------------------------------
## FALSE 154 105 259
## 2.244 2.500
## 0.595 0.405 0.700
## 0.79 0.60
## 0.416 0.284
## -----------------------------------------
## TRUE 41 70 111
## 5.235 5.833
## 0.369 0.631 0.300
## 0.21 0.40
## 0.111 0.189
## -----------------------------------------
## Total 195 175 370
## 0.527 0.473
## =========================================CrossTable(my_data$high_use, my_data$Pstatus)## Cell Contents
## |-------------------------|
## | N |
## | Chi-square contribution |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
## =========================================
## my_data$Pstatus
## my_data$high_use A T Total
## -----------------------------------------
## FALSE 26 233 259
## 0.014 0.002
## 0.100 0.900 0.700
## 0.684 0.702
## 0.070 0.630
## -----------------------------------------
## TRUE 12 99 111
## 0.032 0.004
## 0.108 0.892 0.300
## 0.316 0.298
## 0.032 0.268
## -----------------------------------------
## Total 38 332 370
## 0.103 0.897
## =========================================CrossTable(my_data$high_use, my_data$studytime)## Cell Contents
## |-------------------------|
## | N |
## | Chi-square contribution |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
## =========================================================
## my_data$studytime
## my_data$high_use 1 2 3 4 Total
## ---------------------------------------------------------
## FALSE 56 128 52 23 259
## 2.314 0.017 2.381 0.889
## 0.216 0.494 0.201 0.089 0.700
## 0.571 0.692 0.867 0.852
## 0.151 0.346 0.141 0.062
## ---------------------------------------------------------
## TRUE 42 57 8 4 111
## 5.400 0.041 5.556 2.075
## 0.378 0.514 0.072 0.036 0.300
## 0.429 0.308 0.133 0.148
## 0.114 0.154 0.022 0.011
## ---------------------------------------------------------
## Total 98 185 60 27 370
## 0.265 0.500 0.162 0.073
## =========================================================my_data %>% group_by(high_use) %>% summarise(count = n(), mean_absences = mean(absences))## # A tibble: 2 × 3
## high_use count mean_absences
## <lgl> <int> <dbl>
## 1 FALSE 259 3.71
## 2 TRUE 111 6.38g1 <- ggplot(data = my_data, aes(x = sex))
g1 + geom_bar() + facet_wrap("high_use") 
g2 <- ggplot(data = my_data, aes(x = Pstatus))
g2 + geom_bar() + facet_wrap("high_use") 
g3 <- ggplot(data = my_data, aes(x = studytime))
g3 + geom_bar() + facet_wrap("high_use") 
g4 <- ggplot(data = my_data, aes(x = absences))
g4 + geom_bar() + facet_wrap("high_use") 
Looking at data, it seems that first of all there are more students with low alcohol consumption (259) compared to high- (111). Absences is clearly skewed to left (less absences). It seems that high consumption group has more males (63%) compared to low consumption (41%). Parental status seems to be similarly distributed between the grpoups. In regards to studytime, high consumption group has notably higher frequency of students who study very little (37.8% vs 21.6%). Finally, high consumption group has higher mean absences (6.4), compared to low consumption group (3.7). Taken together, it seems that sex, studytime and absences could maybe predict alcohol consumptions, whereas parental status seems to be pretty evenly distributed between groups and thus will unlikely predict alcohol consumption.
Logistic regression
m <- glm(high_use ~ sex + Pstatus + studytime + absences, data = my_data, family = "binomial")
summary(m)##
## Call:
## glm(formula = high_use ~ sex + Pstatus + studytime + absences,
## family = "binomial", data = my_data)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.1339 -0.8389 -0.5966 1.0728 2.1221
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.01835 0.55916 -1.821 0.068574 .
## sexM 0.84663 0.25509 3.319 0.000903 ***
## PstatusT 0.12379 0.40420 0.306 0.759415
## studytime -0.41526 0.16023 -2.592 0.009550 **
## absences 0.08928 0.02345 3.807 0.000141 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 452.04 on 369 degrees of freedom
## Residual deviance: 408.40 on 365 degrees of freedom
## AIC: 418.4
##
## Number of Fisher Scoring iterations: 4OR <- coef(m) %>% exp
CI <- confint(m) %>% exp## Waiting for profiling to be done...cbind(OR, CI)## OR 2.5 % 97.5 %
## (Intercept) 0.3611909 0.1172496 1.0606103
## sexM 2.3317700 1.4202152 3.8687722
## PstatusT 1.1317735 0.5251790 2.5945605
## studytime 0.6601657 0.4776116 0.8967295
## absences 1.0933826 1.0465506 1.1475274Logistic regression models showed that sex, studytime and absences were significant predictors high/low alcohol consumptions:
sexM log odds .85 (p-value <0.001) studytime log odds -.42 (p-value < 0.01), meaning that for unit increase in studytime the log odds for being n high alcohol consumption decrease by 0.42.
absences log odds .09 (p-value <0.001), meaning that for each absence the log odds for high alcohol consumption changes by 0.09.
The odds for a male being in high alcohol consumption groups over a female is 2.3 (CI 1.4; 3.9). For unit increase in studytime the odds of being in high alcohol consumption group decrese by 34% (CI 0.48; 0.90), note that studytime had four possible values corresponging to <2 hours, 2 to 5 hours, 5 to 10 hours and >10 hours. For each absence the odds of being in high alcohol consumption group increase by 9.3% (CI 4.7%; 14.8%).
Against the initial hypothesis parental status did not predict alcohol consumption, however this was could be predicted already by exploring the data. The effect of sex, studytime and absences were as hypothesized as being a male, lower studytime and higher absences predicted being in high alcohol consumption group.
Predictive power of the model
# drop parental status from the model as it wasn't statistically significant predictor
new_model <- glm(high_use ~ sex + studytime + absences, data = my_data, family = "binomial")
probabilities <- predict(new_model, type = "response")
my_data <- mutate(my_data, probability = probabilities)
my_data <- mutate(my_data, prediction = (probability > 0.5))
table(high_use = my_data$high_use, prediction = my_data$prediction)## prediction
## high_use FALSE TRUE
## FALSE 250 9
## TRUE 86 25loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
# training error
loss_func(class = my_data$high_use, prob = my_data$probability)## [1] 0.2567568g <- ggplot(my_data, aes(x = probability, y = high_use, col = prediction))
g + geom_point()
The model resulted in 250 true negatives, 25 true positives, 86 false negatives and 9 false positives.
The training error of the model is 25.7%.
10-fold cross-validation
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
# compute the average number of wrong predictions in the (training) data
loss_func(class = my_data$high_use, prob = my_data$probability)## [1] 0.2567568# K-fold cross-validation
library(boot)
cv <- cv.glm(data = alc, cost = loss_func, glmfit = m, K = 10)
# average number of wrong predictions in the cross validation
cv$delta[1]## [1] 0.2567568The test set error of my model with 10-fold cross-validation is ~0.27, which is slightly worse than the model introduced in Exercise set.
Chapter 4) Clustering
date()## [1] "Mon Dec 5 09:34:17 2022"Reading data and exploring it
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectlibrary(corrplot)## corrplot 0.92 loadedlibrary(tidyr)
library(ggplot2)
data("Boston")
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...dim(Boston)## [1] 506 14Boston data consists of 14 variables (and 506 observations) and it is about housing values in suburbs of Boston. Details and full descriptions of the variables can be found from Boston.
Graphical overview of the data
#summaries of the variable data
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00pairs(Boston)
#library(Hmisc)
boston_df <- as.data.frame(Boston)
#hist.data.frame(boston_df) this plot gives an error while knitting the index file so unfortunately can't display it
cor_matrix <- cor(Boston) %>% round(digits = 2)
corrplot(cor_matrix, method="circle", type = "upper", cl.pos = "b", tl.pos = "d", tl.cex = 0.6)
The figure of pairs function was so difficult to read on my laptop, so I ended up drawing also histograms to look at the distributions of the variables (unfortunately it can’t be displayed while knitting index because of the plot size, code commented out). Many variables are heavily skewed. The only variables close to normal distribution seem to be ‘average number of rooms per dwelling’(rm) and ‘median value of owner-occupied homes in $1000s’ (medv).
The correlation plot shows the relationships between the variables. Strongest negative correlations can be found between:
* weighted mean of distances to five Boston employment centres (dis) and proportion of owner-occupied units built prior to 1940 (age)
* dis and nitrogen oxides concentration (nox)
* dis and proportion of non-retail business acres per town (indus)
* lower status of the population percent (lastat) and median value of owner-occupied homes in $1000s (medv)
Strongest positive correlation is between index of accessibility to radial highways (rad) and full-value property-tax rate per $10,000 (tax)
Standardize the dataset
boston_scaled <- as.data.frame(scale(Boston))
boston_scaled$crim <- as.numeric(boston_scaled$crim)
# summaries of the scaled variables
summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865# summary of the scaled crime rate
summary(boston_scaled$crim)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.419367 -0.410563 -0.390280 0.000000 0.007389 9.924110# create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
# create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, label = c("low", "med_low", "med_high", "high"))
# look at the table of the new factor crime
table(crime)## crime
## low med_low med_high high
## 127 126 126 127# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
ind <- sample(nrow(boston_scaled), size = nrow(boston_scaled) * 0.8)
train <- boston_scaled[ind,]
test <- boston_scaled[-ind,]
correct_classes <- test$crime
test <- dplyr::select(test, -crime)
boston_scaled$crime <- factor(boston_scaled$crime, levels = c("low", "med_low", "med_high", "high"))Linear discriminant analysis
# linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)
# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "black", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results
plot(lda.fit, dimen = 2, col = classes)
lda.arrows(lda.fit, myscale = 1)
Predict classes with LDA model
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class) ## predicted
## correct low med_low med_high high
## low 19 8 0 0
## med_low 5 18 2 0
## med_high 2 10 15 0
## high 0 0 0 23LDA model seems to predict classes quite well with accuracy of ~75% (varying between seeds). The model performed best in classifying high crime rates.
k-means clustering
# reloading the dataset
library(MASS)
data("Boston")
# standardizing the dataset
boston_scaled2 <- as.data.frame(scale(Boston))
# distances between observations
dist_eu <- dist(boston_scaled2)
# look at the summary of the distances
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970Looking for optimal number of clusters
library(ggplot2)
set.seed(13)
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled2, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
The optimal number of clusters is when the value of total WCSS (y-axis) changes radically, here it seems to be around two clusters.
# k-means clustering
km <- kmeans(boston_scaled2, centers = 2)
# plot the Boston dataset with clusters
pairs(boston_scaled2[1:6], col = km$cluster) 
pairs(boston_scaled2[7:14], col = km$cluster) 
I plotted the clusters in two separate plots, since the figures are otherwise two small for me to see in laptop, but this way I don’t see all the pairs. Overall it looks like two clusters works nicely in most of the pairs.
Super-Bonus:
matrix product
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layout# color is the crime classes of exercise set: classes is determined earlier as classes <- as.numeric(train$crime)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = classes)Chapter 5) Dimensionality reduction techniques
date()## [1] "Mon Dec 5 09:34:23 2022"Overview of the data
The variables of this dataset are:
edu2_fm - Proportion of females with at least secondary education divided by the proportion of males with at least secondary education
labofm - Proportion of females in the labour force divided by the proportion of males in labour force
edu_exp - Expected years of schooling
life_exp - life expectancy
gni - Gross National Income per capita
mat_mor - Maternal mortality ratio
ado_birth - Adolescent birth rate
parli_perc - Percetange of female representatives in parliament
library(tidyr)
library(dplyr)
library(GGally)
library(ggplot2)
library(corrplot)
library(FactoMineR)
human <- read.csv("C:/Users/labpaavo/IODS-project/data/human.csv")
str(human)## 'data.frame': 155 obs. of 8 variables:
## $ edu2_fm : num 1.007 0.997 0.983 0.989 0.969 ...
## $ labofm : num 0.891 0.819 0.825 0.884 0.829 ...
## $ edu_exp : num 17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
## $ life_exp : num 81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
## $ gni : int 64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
## $ mat_mor : int 4 6 6 5 6 7 9 28 11 8 ...
## $ ado_birth : num 7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
## $ parli_perc: num 39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...dim(human)## [1] 155 8summary(human)## edu2_fm labofm edu_exp life_exp
## Min. :0.1717 Min. :0.1857 Min. : 5.40 Min. :49.00
## 1st Qu.:0.7264 1st Qu.:0.5984 1st Qu.:11.25 1st Qu.:66.30
## Median :0.9375 Median :0.7535 Median :13.50 Median :74.20
## Mean :0.8529 Mean :0.7074 Mean :13.18 Mean :71.65
## 3rd Qu.:0.9968 3rd Qu.:0.8535 3rd Qu.:15.20 3rd Qu.:77.25
## Max. :1.4967 Max. :1.0380 Max. :20.20 Max. :83.50
## gni mat_mor ado_birth parli_perc
## Min. : 581 Min. : 1.0 Min. : 0.60 Min. : 0.00
## 1st Qu.: 4198 1st Qu.: 11.5 1st Qu.: 12.65 1st Qu.:12.40
## Median : 12040 Median : 49.0 Median : 33.60 Median :19.30
## Mean : 17628 Mean : 149.1 Mean : 47.16 Mean :20.91
## 3rd Qu.: 24512 3rd Qu.: 190.0 3rd Qu.: 71.95 3rd Qu.:27.95
## Max. :123124 Max. :1100.0 Max. :204.80 Max. :57.50p <- ggpairs(human, lower = list(combo = wrap("facethist", bins = 20)))
p
cor(human) %>% corrplot
From the data we can see that life expectancy is skewed towards right, whereas GNI, maternal mortality ratio and adolescent birth rate are heavily skewed to left. Many variables are strongly correlated with each other, except for gender ratio in labor force (labofm) and gender ratio in parliment.
PCA on raw data
# perform principal component analysis (with the SVD method)
pca_human <- prcomp(human)
biplot(pca_human, choices = 1:2, cex = c(0.6, 1), col = c("grey40", "deeppink2"))## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
summary(pca_human)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8
## Standard deviation 1.854e+04 185.5219 25.19 11.45 3.766 1.566 0.1912 0.1591
## Proportion of Variance 9.999e-01 0.0001 0.00 0.00 0.000 0.000 0.0000 0.0000
## Cumulative Proportion 9.999e-01 1.0000 1.00 1.00 1.000 1.000 1.0000 1.0000We can see that with raw data the 1st principal component accounts for basically all the variation (99,999%). The figure shows that the variable responsible for this is GNI.
PCA on standardized data
human_std <- scale(human)
pca_human_std <- prcomp(human_std)
biplot(pca_human_std, choices = 1:2, cex = c(0.6, 1), col = c("grey40", "deeppink2"))
summary(pca_human_std)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 2.0708 1.1397 0.87505 0.77886 0.66196 0.53631 0.45900
## Proportion of Variance 0.5361 0.1624 0.09571 0.07583 0.05477 0.03595 0.02634
## Cumulative Proportion 0.5361 0.6984 0.79413 0.86996 0.92473 0.96069 0.98702
## PC8
## Standard deviation 0.32224
## Proportion of Variance 0.01298
## Cumulative Proportion 1.00000Now PC1 accounts for ~54% of variation compared to 99.99% previously. This is because the raw values of GNI vary from ~20000 to ~500, whereas i.e., labofm and edu2_fm can only have values between 0 and 1. Thus, if we only use absolute values, GNI will account for practically all the variation in the data. This can be resolved by standardizing the data, which makes the changes in variables more comparable.
Interpretation of the first two PC dimensions
It seems that the in the first two principal component dimensions the variables form three ‘groups’. Maternal mortality ratio and adolescent birth rate to seem to be close to each other, meaning that the two are closely related to each other in this data. Another pair can be seen between gender ratio in labor force and the percentage of females in parliment. The rest of the variables are grouped together.
Tea dataset
tea <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/Helsinki-Open-Data-Science/master/datasets/tea.csv",
sep = ",", header = T)
dim(tea)## [1] 300 36str(tea)## 'data.frame': 300 obs. of 36 variables:
## $ breakfast : chr "breakfast" "breakfast" "Not.breakfast" "Not.breakfast" ...
## $ tea.time : chr "Not.tea time" "Not.tea time" "tea time" "Not.tea time" ...
## $ evening : chr "Not.evening" "Not.evening" "evening" "Not.evening" ...
## $ lunch : chr "Not.lunch" "Not.lunch" "Not.lunch" "Not.lunch" ...
## $ dinner : chr "Not.dinner" "Not.dinner" "dinner" "dinner" ...
## $ always : chr "Not.always" "Not.always" "Not.always" "Not.always" ...
## $ home : chr "home" "home" "home" "home" ...
## $ work : chr "Not.work" "Not.work" "work" "Not.work" ...
## $ tearoom : chr "Not.tearoom" "Not.tearoom" "Not.tearoom" "Not.tearoom" ...
## $ friends : chr "Not.friends" "Not.friends" "friends" "Not.friends" ...
## $ resto : chr "Not.resto" "Not.resto" "resto" "Not.resto" ...
## $ pub : chr "Not.pub" "Not.pub" "Not.pub" "Not.pub" ...
## $ Tea : chr "black" "black" "Earl Grey" "Earl Grey" ...
## $ How : chr "alone" "milk" "alone" "alone" ...
## $ sugar : chr "sugar" "No.sugar" "No.sugar" "sugar" ...
## $ how : chr "tea bag" "tea bag" "tea bag" "tea bag" ...
## $ where : chr "chain store" "chain store" "chain store" "chain store" ...
## $ price : chr "p_unknown" "p_variable" "p_variable" "p_variable" ...
## $ age : int 39 45 47 23 48 21 37 36 40 37 ...
## $ sex : chr "M" "F" "F" "M" ...
## $ SPC : chr "middle" "middle" "other worker" "student" ...
## $ Sport : chr "sportsman" "sportsman" "sportsman" "Not.sportsman" ...
## $ age_Q : chr "35-44" "45-59" "45-59" "15-24" ...
## $ frequency : chr "1/day" "1/day" "+2/day" "1/day" ...
## $ escape.exoticism: chr "Not.escape-exoticism" "escape-exoticism" "Not.escape-exoticism" "escape-exoticism" ...
## $ spirituality : chr "Not.spirituality" "Not.spirituality" "Not.spirituality" "spirituality" ...
## $ healthy : chr "healthy" "healthy" "healthy" "healthy" ...
## $ diuretic : chr "Not.diuretic" "diuretic" "diuretic" "Not.diuretic" ...
## $ friendliness : chr "Not.friendliness" "Not.friendliness" "friendliness" "Not.friendliness" ...
## $ iron.absorption : chr "Not.iron absorption" "Not.iron absorption" "Not.iron absorption" "Not.iron absorption" ...
## $ feminine : chr "Not.feminine" "Not.feminine" "Not.feminine" "Not.feminine" ...
## $ sophisticated : chr "Not.sophisticated" "Not.sophisticated" "Not.sophisticated" "sophisticated" ...
## $ slimming : chr "No.slimming" "No.slimming" "No.slimming" "No.slimming" ...
## $ exciting : chr "No.exciting" "exciting" "No.exciting" "No.exciting" ...
## $ relaxing : chr "No.relaxing" "No.relaxing" "relaxing" "relaxing" ...
## $ effect.on.health: chr "No.effect on health" "No.effect on health" "No.effect on health" "No.effect on health" ...#View(tea)
ind <- 1:ncol(tea)
tea[, ind] <- lapply(tea[, ind], as.factor)
str(tea)## 'data.frame': 300 obs. of 36 variables:
## $ breakfast : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
## $ tea.time : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
## $ evening : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
## $ lunch : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
## $ dinner : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
## $ always : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
## $ home : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
## $ work : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
## $ tearoom : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ friends : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
## $ resto : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
## $ pub : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ sugar : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ where : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ price : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
## $ age : Factor w/ 61 levels "15","17","18",..: 24 30 32 8 33 6 22 21 25 22 ...
## $ sex : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
## $ SPC : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
## $ Sport : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
## $ age_Q : Factor w/ 5 levels "+60","15-24",..: 4 5 5 2 5 2 4 4 4 4 ...
## $ frequency : Factor w/ 4 levels "+2/day","1 to 2/week",..: 3 3 1 3 1 3 4 2 1 1 ...
## $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
## $ spirituality : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
## $ healthy : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
## $ diuretic : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
## $ friendliness : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
## $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ feminine : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
## $ sophisticated : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
## $ slimming : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ exciting : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
## $ relaxing : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
## $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...library(Hmisc)## Loading required package: lattice##
## Attaching package: 'lattice'## The following object is masked from 'package:boot':
##
## melanoma## Loading required package: survival##
## Attaching package: 'survival'## The following object is masked from 'package:boot':
##
## aml## Loading required package: Formula##
## Attaching package: 'Hmisc'## The following object is masked from 'package:plotly':
##
## subplot## The following objects are masked from 'package:dplyr':
##
## src, summarize## The following objects are masked from 'package:base':
##
## format.pval, unitshist.data.frame(tea[1:9])
hist.data.frame(tea[10:18])
hist.data.frame(tea[19:28])
hist.data.frame(tea[29:36])
MCA on tea data
The categorical variables chosen for the MCA were: Tea (black, earl grey, green), How (alone, lemon, milk, other), how (tea bag, tea bag + unpackagerd, unpackaged), sugar (no sugar, sugar), where (chain store, tea shop, chain store + tea shop) and lunch (lunch, not lunch).
# select some columns
keep_columns <- c("Tea", "How", "how", "sugar", "where", "lunch")
tea_time <- select(tea, keep_columns)## Warning: Using an external vector in selections was deprecated in tidyselect 1.1.0.
## ℹ Please use `all_of()` or `any_of()` instead.
## # Was:
## data %>% select(keep_columns)
##
## # Now:
## data %>% select(all_of(keep_columns))
##
## See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.mca <- MCA(tea_time, graph = FALSE)
# summary of the model
summary(mca)##
## Call:
## MCA(X = tea_time, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6 Dim.7
## Variance 0.279 0.261 0.219 0.189 0.177 0.156 0.144
## % of var. 15.238 14.232 11.964 10.333 9.667 8.519 7.841
## Cumulative % of var. 15.238 29.471 41.435 51.768 61.434 69.953 77.794
## Dim.8 Dim.9 Dim.10 Dim.11
## Variance 0.141 0.117 0.087 0.062
## % of var. 7.705 6.392 4.724 3.385
## Cumulative % of var. 85.500 91.891 96.615 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.298 0.106 0.086 | -0.328 0.137 0.105 | -0.327
## 2 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 3 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 4 | -0.530 0.335 0.460 | -0.318 0.129 0.166 | 0.211
## 5 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 6 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 7 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 8 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 9 | 0.143 0.024 0.012 | 0.871 0.969 0.435 | -0.067
## 10 | 0.476 0.271 0.140 | 0.687 0.604 0.291 | -0.650
## ctr cos2
## 1 0.163 0.104 |
## 2 0.735 0.314 |
## 3 0.062 0.069 |
## 4 0.068 0.073 |
## 5 0.062 0.069 |
## 6 0.062 0.069 |
## 7 0.062 0.069 |
## 8 0.735 0.314 |
## 9 0.007 0.003 |
## 10 0.643 0.261 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr cos2
## black | 0.473 3.288 0.073 4.677 | 0.094 0.139 0.003
## Earl Grey | -0.264 2.680 0.126 -6.137 | 0.123 0.626 0.027
## green | 0.486 1.547 0.029 2.952 | -0.933 6.111 0.107
## alone | -0.018 0.012 0.001 -0.418 | -0.262 2.841 0.127
## lemon | 0.669 2.938 0.055 4.068 | 0.531 1.979 0.035
## milk | -0.337 1.420 0.030 -3.002 | 0.272 0.990 0.020
## other | 0.288 0.148 0.003 0.876 | 1.820 6.347 0.102
## tea bag | -0.608 12.499 0.483 -12.023 | -0.351 4.459 0.161
## tea bag+unpackaged | 0.350 2.289 0.056 4.088 | 1.024 20.968 0.478
## unpackaged | 1.958 27.432 0.523 12.499 | -1.015 7.898 0.141
## v.test Dim.3 ctr cos2 v.test
## black 0.929 | -1.081 21.888 0.382 -10.692 |
## Earl Grey 2.867 | 0.433 9.160 0.338 10.053 |
## green -5.669 | -0.108 0.098 0.001 -0.659 |
## alone -6.164 | -0.113 0.627 0.024 -2.655 |
## lemon 3.226 | 1.329 14.771 0.218 8.081 |
## milk 2.422 | 0.013 0.003 0.000 0.116 |
## other 5.534 | -2.524 14.526 0.197 -7.676 |
## tea bag -6.941 | -0.065 0.183 0.006 -1.287 |
## tea bag+unpackaged 11.956 | 0.019 0.009 0.000 0.226 |
## unpackaged -6.482 | 0.257 0.602 0.009 1.640 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## Tea | 0.126 0.108 0.410 |
## How | 0.076 0.190 0.394 |
## how | 0.708 0.522 0.010 |
## sugar | 0.065 0.001 0.336 |
## where | 0.702 0.681 0.055 |
## lunch | 0.000 0.064 0.111 |# visualize MCA
plot(mca, invisible=c("ind"), graph.type = "classic", habillage = "quali")
plot(mca, invisible=c("var"), graph.type = "classic")
The MCA factor map shows gactors that are close to each other in the data. We can see that tea bought at tea shop is usually unpackaged. Another group can be seen with variables that describe incosistent behavior: buying tea from both chain stores and tea shops is related to having both tea bags and unpackaged teas as well as non-consistent way of drinking tea (other). The other figure shows the individuals in the MCA factor map.